Word problems with rational numbers
In grade 7 we stop pretending that numbers cannot be negative. Frost, debt, depth — they are all stories where signed numbers help. The agreement is simple:
- rise, gain, deposit: add a positive number
- fall, expense, depth: add a negative number
Type 1: temperature change
Problem. At sunset it was 5 °C. Overnight the temperature fell by 8 °C and by morning it had risen by 3 °C. What is the morning temperature?
Working:
`5 + (−8) + 3 = 5 − 8 + 3 = 0 °C`
Sanity check: the drop is bigger than the starting value, so the answer will be close to zero or negative.
Type 2: account balance
Problem. Tom had £30 on his card. He spent £50 in the bookshop, then a £20 refund was credited. What is his balance?
Working:
`30 + (−50) + 20 = £0`
Without the refund the balance would have been `30 − 50 = −£20`, which means an overdraft of £20.
Type 3: elevation change (decimal)
Problem. A diver was at a depth of 6 m (an elevation of −6 m). She rises 3.5 m and then descends another 2 m. What is her new elevation?
Working:
`−6 + 3.5 + (−2) = −4.5 m`
The negative sign tells you she is still underwater.
Tips and traps
- First write each change as a signed number, then add.
- For longer chains, write the whole calculation in one line: `+5 − 8 + 3`. That stops sign slips.
- Sanity-check the final sign and unit.